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from functools import singledispatch |
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from sympy.core.numbers import pi |
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from sympy.functions.elementary.trigonometric import tan |
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from sympy.simplify import trigsimp |
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from sympy.core import Basic, Tuple |
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from sympy.core.symbol import _symbol |
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from sympy.solvers import solve |
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from sympy.geometry import Point, Segment, Curve, Ellipse, Polygon |
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from sympy.vector import ImplicitRegion |
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class ParametricRegion(Basic): |
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""" |
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Represents a parametric region in space. |
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Examples |
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======== |
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>>> from sympy import cos, sin, pi |
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>>> from sympy.abc import r, theta, t, a, b, x, y |
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>>> from sympy.vector import ParametricRegion |
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>>> ParametricRegion((t, t**2), (t, -1, 2)) |
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ParametricRegion((t, t**2), (t, -1, 2)) |
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>>> ParametricRegion((x, y), (x, 3, 4), (y, 5, 6)) |
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ParametricRegion((x, y), (x, 3, 4), (y, 5, 6)) |
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>>> ParametricRegion((r*cos(theta), r*sin(theta)), (r, -2, 2), (theta, 0, pi)) |
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ParametricRegion((r*cos(theta), r*sin(theta)), (r, -2, 2), (theta, 0, pi)) |
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>>> ParametricRegion((a*cos(t), b*sin(t)), t) |
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ParametricRegion((a*cos(t), b*sin(t)), t) |
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>>> circle = ParametricRegion((r*cos(theta), r*sin(theta)), r, (theta, 0, pi)) |
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>>> circle.parameters |
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(r, theta) |
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>>> circle.definition |
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(r*cos(theta), r*sin(theta)) |
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>>> circle.limits |
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{theta: (0, pi)} |
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Dimension of a parametric region determines whether a region is a curve, surface |
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or volume region. It does not represent its dimensions in space. |
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>>> circle.dimensions |
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1 |
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Parameters |
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========== |
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definition : tuple to define base scalars in terms of parameters. |
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bounds : Parameter or a tuple of length 3 to define parameter and corresponding lower and upper bound. |
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""" |
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def __new__(cls, definition, *bounds): |
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parameters = () |
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limits = {} |
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if not isinstance(bounds, Tuple): |
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bounds = Tuple(*bounds) |
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for bound in bounds: |
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if isinstance(bound, (tuple, Tuple)): |
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if len(bound) != 3: |
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raise ValueError("Tuple should be in the form (parameter, lowerbound, upperbound)") |
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parameters += (bound[0],) |
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limits[bound[0]] = (bound[1], bound[2]) |
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else: |
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parameters += (bound,) |
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if not isinstance(definition, (tuple, Tuple)): |
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definition = (definition,) |
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obj = super().__new__(cls, Tuple(*definition), *bounds) |
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obj._parameters = parameters |
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obj._limits = limits |
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return obj |
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@property |
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def definition(self): |
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return self.args[0] |
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@property |
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def limits(self): |
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return self._limits |
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@property |
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def parameters(self): |
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return self._parameters |
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@property |
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def dimensions(self): |
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return len(self.limits) |
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@singledispatch |
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def parametric_region_list(reg): |
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""" |
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Returns a list of ParametricRegion objects representing the geometric region. |
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Examples |
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======== |
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>>> from sympy.abc import t |
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>>> from sympy.vector import parametric_region_list |
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>>> from sympy.geometry import Point, Curve, Ellipse, Segment, Polygon |
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>>> p = Point(2, 5) |
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>>> parametric_region_list(p) |
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[ParametricRegion((2, 5))] |
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>>> c = Curve((t**3, 4*t), (t, -3, 4)) |
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>>> parametric_region_list(c) |
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[ParametricRegion((t**3, 4*t), (t, -3, 4))] |
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>>> e = Ellipse(Point(1, 3), 2, 3) |
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>>> parametric_region_list(e) |
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[ParametricRegion((2*cos(t) + 1, 3*sin(t) + 3), (t, 0, 2*pi))] |
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>>> s = Segment(Point(1, 3), Point(2, 6)) |
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>>> parametric_region_list(s) |
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[ParametricRegion((t + 1, 3*t + 3), (t, 0, 1))] |
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>>> p1, p2, p3, p4 = [(0, 1), (2, -3), (5, 3), (-2, 3)] |
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>>> poly = Polygon(p1, p2, p3, p4) |
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>>> parametric_region_list(poly) |
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[ParametricRegion((2*t, 1 - 4*t), (t, 0, 1)), ParametricRegion((3*t + 2, 6*t - 3), (t, 0, 1)),\ |
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ParametricRegion((5 - 7*t, 3), (t, 0, 1)), ParametricRegion((2*t - 2, 3 - 2*t), (t, 0, 1))] |
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""" |
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raise ValueError("SymPy cannot determine parametric representation of the region.") |
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@parametric_region_list.register(Point) |
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def _(obj): |
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return [ParametricRegion(obj.args)] |
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@parametric_region_list.register(Curve) |
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def _(obj): |
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definition = obj.arbitrary_point(obj.parameter).args |
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bounds = obj.limits |
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return [ParametricRegion(definition, bounds)] |
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@parametric_region_list.register(Ellipse) |
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def _(obj, parameter='t'): |
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definition = obj.arbitrary_point(parameter).args |
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t = _symbol(parameter, real=True) |
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bounds = (t, 0, 2*pi) |
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return [ParametricRegion(definition, bounds)] |
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@parametric_region_list.register(Segment) |
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def _(obj, parameter='t'): |
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t = _symbol(parameter, real=True) |
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definition = obj.arbitrary_point(t).args |
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for i in range(0, 3): |
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lower_bound = solve(definition[i] - obj.points[0].args[i], t) |
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upper_bound = solve(definition[i] - obj.points[1].args[i], t) |
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if len(lower_bound) == 1 and len(upper_bound) == 1: |
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bounds = t, lower_bound[0], upper_bound[0] |
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break |
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definition_tuple = obj.arbitrary_point(parameter).args |
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return [ParametricRegion(definition_tuple, bounds)] |
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@parametric_region_list.register(Polygon) |
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def _(obj, parameter='t'): |
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l = [parametric_region_list(side, parameter)[0] for side in obj.sides] |
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return l |
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@parametric_region_list.register(ImplicitRegion) |
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def _(obj, parameters=('t', 's')): |
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definition = obj.rational_parametrization(parameters) |
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bounds = [] |
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for i in range(len(obj.variables) - 1): |
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parameter = _symbol(parameters[i], real=True) |
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definition = [trigsimp(elem.subs(parameter, tan(parameter/2))) for elem in definition] |
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bounds.append((parameter, 0, 2*pi),) |
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definition = Tuple(*definition) |
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return [ParametricRegion(definition, *bounds)] |
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